We establish a criterion for a semigroup identity to hold in the monoid of \(n \times n\) upper unitriangular matrices with entries in a commutative semiring S. This criterion is combinatorial modulo the arithmetic of the multiplicative identity element of S. In the case where S is non-trivial and idempotent, the generated variety is the variety \(\mathbf {J}_{\mathbf {n-1...

In this paper we initiate the study of \(\aleph _0\)-categorical semigroups, where a countable semigroup S is \(\aleph _0\)-categorical if, for any natural number n, the action of its group of automorphisms \({\text {Aut}}(S)\) on \(S^n\) has only finitely many orbits. We show that \(\aleph _0\)-categoricity transfers to certain important substructures such as maximal subgroups...

A smallest generating set of a semigroup is a generating set of the smallest cardinality. Similarly, an irredundant generating setX is a generating set such that no proper subset of X is also a generating set. A semigroup S is ubiquitous if every irredundant generating set of S is of the same cardinality. We are motivated by a naïve algorithm to find a small generating set for a...

We investigate the preservation of the properties of being finitely generated and finitely presented under both direct and wreath products of monoid acts. A monoid M is said to preserve property \({\mathcal {P}}\) in direct products if, for any two M-acts A and B, the direct product \(A\times B\) has property \({\mathcal {P}}\) if and only if both A and B have property...

A totally ordered monoid, or tomonoid for short, is a monoid endowed with a compatible total order. We reconsider in this paper the problem of describing the one-element Rees coextensions of a finite, negative tomonoid S, that is, those tomonoids that are by one element larger than S and whose Rees quotient by the poideal consisting of the two smallest elements is isomorphic to S...

The variety \({\mathcal Z}\) of commutative additively and multiplicatively idempotent semirings is studied. We prove that \({\mathcal Z}\) is generated by a single subdirectly irreducible three-element semiring and it has a canonical form for its terms. Hence, \({\mathcal Z}\) is locally finite despite the fact that it is residually large. The word problem in \({\mathcal Z}\) is...

In the lattice of subvarieties of the variety \({\mathcal {V}= [xyz=xywyz]}\) depicted in p. 370, the following inclusions are missing.

A right chain ordered semigroup is an ordered semigroup whose right ideals form a chain. In this paper we study the ideal theory of right chain ordered semigroups in terms of prime ideals, completely prime ideals and prime segments, extending to these semigroups results on right chain semigroups proved in Ferrero et al. (J Algebra 292:574–584, 2005).

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for...

Starting from the symmetric group \(S_n\), we construct two fiat 2-categories. One of them can be viewed as the fiat “extension” of the natural 2-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the symmetric...

In this paper we consider the Schwarz radical of linear algebraic semigroups as defined in semigroup theory. We give some new characterizations of the complete regularity, regularity and solvability of irreducible linear algebraic monoids in terms of Schwarz radical data. Moreover, we give a generalization about the results of the kernel to the results of completely regular...

This paper studies semigroups of operators on Hardy and Dirichlet spaces whose generators are differential operators of order greater than one. The theory of forms is used to provide conditions for the generation of semigroups by second order differential operators. Finally, a class of more general weighted Hardy spaces is considered and necessary and sufficient conditions are...

We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf’s Theorem, stating that an infinite group has 1, 2 or infinitely many...

We study maximal subgroups of the free idempotent generated semigroup \({\text {IG}}(E),\) where E is the biordered set of idempotents of the endomorphism monoid \({\text {End}}\mathbf {A}\) of an independence algebra \(\mathbf {A}\), in the case where \(\mathbf {A}\) has no constants and has finite rank n. It is shown that when \(n\ge 3\) the maximal subgroup of \({\text {IG}}(E...